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Accurate quantification of cerebral blood flow (CBF) is essential for the diagnosis and assessment of cerebrovascular diseases such as Moyamoya, carotid stenosis, aneurysms, and stroke. Positron emission tomography (PET) is currently regarded as the gold standard for the measurement of CBF in the human brain. PET imaging, however, is not widely available because of its prohibitive costs, use of ionizing radiation, and logistical challenges, which require a co-localized cyclotron to deliver the 2 min half-life Oxygen-15 radioisotope. Magnetic resonance imaging (MRI), in contrast, is more readily available and does not involve ionizing radiation. In this study, we propose a multi-task learning framework for brain MRI-to-PET translation and disease diagnosis. The proposed framework comprises two prime networks: (1) an attention-based 3D encoder-decoder convolutional neural network (CNN) that synthesizes high-quality PET CBF maps from multi-contrast MRI images, and (2) a multi-scale 3D CNN that identifies the brain disease corresponding to the input MRI images. Our multi-task framework yields promising results on the task of MRI-to-PET translation, achieving an average structural similarity index (SSIM) of 0.94 and peak signal-to-noise ratio (PSNR) of 38dB on a cohort of 120 subjects. In addition, we show that integrating multiple MRI modalities can improve the clinical diagnosis of brain diseases.